Download Marcos Vielman Journal chap. 5

 A line that bisects a segment and is
perpendicular to that segment.
 Any point that lies on the perpendicular
bisector, is equidistant to both of the
endpoints of the segment.
 If it is equidistant from both of the endpoints
of the segment, then it is on the
perpendicular bisector.
Examples of Perpendicular
Bisector
B
A
E
D
C
BE = DE
AE = CE
m<E = 90
 A angle bisector is a ray that cuts an angle
into 2 congruent angles. It always lies on the
interior of the angle.
 Any point that lies on the angle bisector is
equidistant to both of the sides of the angle.
 If it is equidistant, then it lies on the angle
bisector.
Examples of Angle bisector
A
D
B
C
AB = BC
ADB = CDB
M AB = M BC
 Concurrent is when three or more lines
intersect at one point.
 The concurrency of the perpendicular
bisector is the circumcenter because it is
where 3 lines meet.

Circumcenter is the point of concurrency
where the perpendicular bisectors of a
triangle meet. It is equidistant to the 3
vertices.
Examples of concurrency
A voting post would be an
example of circumcenter
because it is equidistant to all
vertices so a voting post
needs to be equidistant to 3
towns
 The incenter is the point of concurrency of
the angle bisector because it is where 3 lines
meet.
 Incenter is the point where the angle
bisector of a triangle intersect. It is
equidistant to the sides of the triangle.
Examples of Incenter
A restaurant in the
Middle of 3 highways
Is an example of incenter
Because it is equidistant to
The 3 sides or highways
 Median is the segment that goes from the
vertex of a triangle to the opposite
midpoint.
 Centroid is the point where the medians of a
triangle intersect.
 When the median goes from the vertex to
the opposite midpoint you can see that it
makes to congruent parts so one side is
concurrent to the other.
Examples of Median
The cockpit of a jet
Would be an example of
Centroid because it is the
Center of balance.
 The altitude of a triangle is a segment that
goes from the vertex perpendicular to the
line containing the opposite side.
 The orthocenter is where the altitudes
intersect.
 The concurrency of the altitudes is the
orthocenter because it is where 3 lines
meet.
Altitude
 Is a point that joins two midpoints of the
sides of the triangles.
 The midsegment is parallel to the opposite
side and the midsegment is half as long as
the opposite side.
Midsegment
 In any triangle the longest side is always
opposite the biggest angle, the shortest side
is opposite the shortest angle.
Triangle side angle
relationship
B
X
Y
Z
C
A
AB > BC, m of angle C >
m of angle A
m< Z < m< Y,
XY>XZ
E
DE > EF, m< D
< m< F
F
D
 The measure of an exterior angle of a
triangle is equal to the sum of the measures
of its remote interior angles
Exterior angel inequality
M<4= m<1+m<2
2
3
1
4
45
90
60
60
M<4 = 120
60
M<4= 135
45
 The two smaller sides of a triangle must add
up to more than the length at the length of
the 3rd side.
Triangle inequality
A
C
B
AB+BC > AC
BC+AC > AB
AC+AB > BC
 Indirect proofs are proofs that you use to
proof something that is not right by
contradicting yourself at one point.
1. First you assume what you are proving is
false.
2. Second use that as your given and start
proving
3. Last you find a contradiction and prove it.
Indirect proofs
Statement
1.FH is a median
of triangle DFG
M<DHF > M< GHF
Reason
Statement
GIVEN
1. A triangle has
Two right angles
<1+<2
2. DH ≅ GH
Definition of median
3. FH ≅ HF
Reflexive property
4. DF > GF
Hinge Theorem
2. M<1=m<2=90
3. m<1+m<2 =180
4.M<1+m<2 +m<3=180
5. m<3 =o
Statement
Reason
1. a>0 so 1/a<0
Given
2. 1/a<0
3. 1<0
Therefore if a>0 1/a>0
Given
Multiplicative prop.
Reason
GIVEN
Def. right <
Substitution
Triangle sum
Theorem
A triangle
Can’t have
2 right <‘s
 If two triangles have two sides that are
congruent, but the third side is not
congruent, then the triangle with the larger
included angle has the longer third side.
 If the triangle with the larger included angle
has the longer third side, but the third side is
not congruent, then two triangles have two
sides that are congruent.
Hinge theorem
E
B
m<A > m<D
BC > EF
D
F
C
A
6
L
57
KL<MN
6
m<PQS > m<RQS
7
53
K
Q
N
P
5.3
7
S
5.1
R
 In a 45-45-90 triangle, both legs are
congruent, and the length of the
hypotenuse is the length of a leg times √2
 In a 30-60-90 triangle, the hypotenuse is
twice as long as the shorter leg and the
length of the longer leg is the length of the
shorter leg times √3
Triangles 45-45-90, 30-60-90
45
l√2
x
45
l
X= 7√2
45
7
l
30
30
2s
16
s√3
y
60
s
60
x
16=2x
8=x
y=x√3
y=8√3